How to Put Rainbow Sprinkles on a Donut: Chromatic Numbers of Abelian Cayley Graphs. Also: The Master’s in Mathematics Program at Cal State LA.

Here’s a puzzle. Suppose a professor wants to make tests for her students. She wants to make different versions, so that students sitting near each other cannot copy off of one another. Her class has 15 students sitting in 15 seats around a circular table. Whenever students are sitting one seat or two seats apart from each other, she wants them to have different versions. However, to avoid having to do too much work, she wants to make as few different versions as possible. How many versions of the test should she make? You can imagine variations of this puzzle. For example, what if her class had 13 students, or 20 students, or n students? What if instead of students sitting one or two seats apart getting different versions, students sitting one or five seats apart get different versions? (Professors can be eccentric, after all.) In this talk, we explore these types of questions. We can visualize a situation like this by, of all things, placing rainbow sprinkles on a donut. In a more generalized framing of the problem (i.e, determining chromatic numbers of abelian Cayley graphs), we can encode the relevant information in a matrix, then use linear algebra to find a solution in many cases. This general problem has been studied using ad hoc methods in disparate contexts. We will discuss how our methods provide a unifying framework for this class of problems. In addition to talking about this research, a few minutes of this presentation will be spent talking about the Master’s in Mathematics Program at Cal State LA. This program can serve as a stepping stone from a bachelor’s degree to a PhD in mathematics, particularly for students who would like to solidify their preparation for advanced coursework and perhaps even have the opportunity to try their hand at research.